fluxhelper.hh

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#ifndef DUMUX_NAVIERSTOKES_MOMENTUM_BOUNDARY_FLUXHELPER_HH
#define DUMUX_NAVIERSTOKES_MOMENTUM_BOUNDARY_FLUXHELPER_HH
 
#include <dumux/common/math.hh>
#include <dumux/common/parameters.hh>
#include <dumux/discretization/method.hh>
#include <dumux/freeflow/navierstokes/momentum/velocitygradients.hh>
 
namespace Dumux {

struct NavierStokesMomentumBoundaryFluxHelper
{
    template<class Problem, class FVElementGeometry, class ElementVolumeVariables, class ElementFluxVariablesCache, class Scalar>
    static auto fixedPressureMomentumFlux(const Problem& problem,
                                          const FVElementGeometry& fvGeometry,
                                          const typename FVElementGeometry::SubControlVolumeFace& scvf,
                                          const ElementVolumeVariables& elemVolVars,
                                          const ElementFluxVariablesCache& elemFluxVarsCache,
                                          const Scalar pressure,
                                          const bool zeroNormalVelocityGradient = true)
    {
        // TODO density upwinding?
        static_assert(FVElementGeometry::GridGeometry::discMethod == DiscretizationMethods::fcstaggered);
        using MomentumFlux = typename Problem::MomentumFluxType;
        constexpr std::size_t dim = static_cast<std::size_t>(FVElementGeometry::GridGeometry::GridView::dimension);
        static_assert(
            MomentumFlux::size() == dim,
            "Expects momentum flux to have as many entries as dimensions."
        );
 
        const auto& element = fvGeometry.element();
 
        static_assert(
            std::decay_t<decltype(problem.dirichlet(element, scvf))>::size() == dim,
            "Expects problem.dirichlet to return an array with as many entries as dimensions."
        );
 
        if (scvf.isFrontal())
        {
            MomentumFlux flux(0.0);
 
            // pressure contribution
            flux[scvf.normalAxis()] = (pressure - problem.referencePressure(element, fvGeometry, scvf)) * scvf.directionSign();
 
            // advective terms
            if (problem.enableInertiaTerms())
            {
                const auto v = elemVolVars[scvf.insideScvIdx()].velocity();
                flux[scvf.normalAxis()] += v*v * problem.density(element, fvGeometry, scvf) * scvf.directionSign();
            }
 
            return flux;
        }
        else if (scvf.isLateral())
        {
            // if no zero velocity gradient is desired the lateral flux is zero
            if (!zeroNormalVelocityGradient)
                return MomentumFlux(0.0);
 
            // otherwise, make sure the flow does not diverge by accounting for the off-diagonal entries of the stress tensor
 
            // if the lateral scvf is adjacent to a slip boundary, call the helper function for this special case
            const auto& scv = fvGeometry.scv(scvf.insideScvIdx());
            if (scv.boundary() && problem.onSlipBoundary(fvGeometry, fvGeometry.frontalScvfOnBoundary(scv)))
                return slipVelocityMomentumFlux(problem, fvGeometry, scvf, elemVolVars, elemFluxVarsCache);
 
            // viscous terms
            MomentumFlux flux(0.0);
            const Scalar mu = problem.effectiveViscosity(element, fvGeometry, scvf);
 
            // lateral face normal to boundary (integration point touches boundary)
            if (scv.boundary())
                flux[scv.dofAxis()] -= mu * StaggeredVelocityGradients::velocityGradIJ(fvGeometry, scvf, elemVolVars)
                                               * scvf.directionSign();
            // lateral face coinciding with boundary
            else if (scvf.boundary())
                flux[scv.dofAxis()] -= mu * StaggeredVelocityGradients::velocityGradJI(fvGeometry, scvf, elemVolVars)
                                              * scvf.directionSign();
 
            // advective terms
            if (problem.enableInertiaTerms()) // TODO revise advection terms! Take care with upwinding!
            {
                const auto transportingVelocity = [&]()
                {
                    const auto& orthogonalScvf = fvGeometry.lateralOrthogonalScvf(scvf);
                    const auto innerTransportingVelocity = elemVolVars[orthogonalScvf.insideScvIdx()].velocity();
 
                    static const bool useOldScheme = getParam<bool>("FreeFlow.UseOldTransportingVelocity", true); // TODO how to deprecate?
 
                    if (useOldScheme)
                    {
                        if (scvf.boundary())
                        {
                            if (problem.boundaryTypes(element, scvf).isDirichlet(scvf.normalAxis()))
                                return problem.dirichlet(element, scvf)[scvf.normalAxis()];
                        }
                        else
                            return innerTransportingVelocity;
                    }
 
                    // use the Dirichlet velocity as transporting velocity if the lateral face is on a Dirichlet boundary
                    if (scvf.boundary())
                    {
                        if (problem.boundaryTypes(element, scvf).isDirichlet(scvf.normalAxis()))
                            return 0.5*(problem.dirichlet(element, scvf)[scvf.normalAxis()] + innerTransportingVelocity);
                    }
 
                    if (orthogonalScvf.boundary())
                    {
                        if (problem.boundaryTypes(element, orthogonalScvf).isDirichlet(scvf.normalAxis()))
                            return 0.5*(problem.dirichlet(element, scvf)[scvf.normalAxis()] + innerTransportingVelocity);
                        else
                            return innerTransportingVelocity; // fallback value, should actually never be called
                    }
 
                    // average the transporting velocity by weighting with the scv volumes
                    const auto insideVolume = fvGeometry.scv(orthogonalScvf.insideScvIdx()).volume();
                    const auto outsideVolume = fvGeometry.scv(orthogonalScvf.outsideScvIdx()).volume();
                    const auto outerTransportingVelocity = elemVolVars[orthogonalScvf.outsideScvIdx()].velocity();
                    return (insideVolume*innerTransportingVelocity + outsideVolume*outerTransportingVelocity) / (insideVolume + outsideVolume);
                }();
 
                // lateral face normal to boundary (integration point touches boundary)
                if (scv.boundary())
                {
                    const auto innerVelocity = elemVolVars[scvf.insideScvIdx()].velocity();
                    const auto outerVelocity = elemVolVars[scvf.outsideScvIdx()].velocity();
 
                    const auto rho = problem.insideAndOutsideDensity(element, fvGeometry, scvf);
                    const bool selfIsUpstream = scvf.directionSign() == sign(transportingVelocity);
 
                    const auto insideMomentum = innerVelocity * rho.first;
                    const auto outsideMomentum = outerVelocity * rho.second;
 
                    static const auto upwindWeight = getParamFromGroup<Scalar>(problem.paramGroup(), "Flux.UpwindWeight");
 
                    const auto transportedMomentum =  selfIsUpstream ? (upwindWeight * insideMomentum + (1.0 - upwindWeight) * outsideMomentum)
                                                                     : (upwindWeight * outsideMomentum + (1.0 - upwindWeight) * insideMomentum);
 
                    flux[scv.dofAxis()] += transportingVelocity * transportedMomentum * scvf.directionSign();
                }
 
                // lateral face coinciding with boundary
                else if (scvf.boundary())
                {
                    const auto insideDensity = problem.density(element, fvGeometry.scv(scvf.insideScvIdx()));
                    const auto innerVelocity = elemVolVars[scvf.insideScvIdx()].velocity();
                    flux[scv.dofAxis()] += innerVelocity * transportingVelocity * insideDensity * scvf.directionSign();
                }
            }
 
            return flux;
        }
 
        DUNE_THROW(Dune::InvalidStateException, "Face is neither lateral nor frontal.");
    }

    template<class Problem, class FVElementGeometry, class ElementVolumeVariables, class ElementFluxVariablesCache, class TangentialVelocityGradient = double>
    static auto slipVelocityMomentumFlux(const Problem& problem,
                                         const FVElementGeometry& fvGeometry,
                                         const typename FVElementGeometry::SubControlVolumeFace& scvf,
                                         const ElementVolumeVariables& elemVolVars,
                                         const ElementFluxVariablesCache& elemFluxVarsCache,
                                         const TangentialVelocityGradient& tangentialVelocityGradient = TangentialVelocityGradient(0.0))
    {
        static_assert(FVElementGeometry::GridGeometry::discMethod == DiscretizationMethods::fcstaggered);
        using MomentumFlux = typename Problem::MomentumFluxType;
        constexpr std::size_t dim = static_cast<std::size_t>(FVElementGeometry::GridGeometry::GridView::dimension);
        static_assert(
            MomentumFlux::size() == dim,
            "Expects momentum flux to have as many entries as dimensions."
        );
 
        MomentumFlux flux(0.0);
 
        // slip velocity momentum contribution only makes sense for lateral scvfs
        if (scvf.isFrontal())
            return flux;
 
        using Scalar = std::decay_t<decltype(flux[0])>;
        const auto& scv = fvGeometry.scv(scvf.insideScvIdx());
        const auto& orthogonalScvf = fvGeometry.lateralOrthogonalScvf(scvf);
        const auto& orthogonalScv = fvGeometry.scv(orthogonalScvf.insideScvIdx());
 
        if (scvf.boundary() && problem.onSlipBoundary(fvGeometry, fvGeometry.frontalScvfOnBoundary(orthogonalScv)))
        {
            /*
            *                     ---------#######:::::::::     x dof position   ***** porous boundary at bottom
            *       slip          |      ||      |       ::
            *       gradient      |      ||      |  velI ::     -- element
            *       ------->      |      || scv  x~~~~>  ::
            *       ------>       |      ||      |       ::     O position at which gradient is evaluated (integration point)
            *       ----->        | velJ ^       |       :^
            *       ---->         -------|-######O::::::::| <----This velocity dof (outer velJ) does not exist if the scv itself lies on a
            *                     ***************~~~>******     non-Dirichlet boundary. In that case, use the given tangentialVelocityGradient.
            *                      frontal scvf  v_slip
            *                      on porous                    || and # staggered half-control-volume (own element)
            *                      boundary
            *                                                   :: staggered half-control-volume (neighbor element)
            *
            */
            const Scalar velI = elemVolVars[scvf.insideScvIdx()].velocity();
 
            // viscous terms
            const Scalar mu = problem.effectiveViscosity(fvGeometry.element(), fvGeometry, scvf);
            const Scalar distance = (scv.dofPosition()- scvf.ipGlobal()).two_norm();
            const Scalar velGradJI = [&]
            {
                if (elemVolVars.hasVolVars(orthogonalScvf.outsideScvIdx()))
                    return StaggeredVelocityGradients::velocityGradJI(fvGeometry, scvf, elemVolVars);
                else
                    return tangentialVelocityGradient;
            }();
 
            const Scalar slipVelocity = problem.beaversJosephVelocity(fvGeometry, scvf, elemVolVars, velGradJI)[scv.dofAxis()]; // TODO rename to slipVelocity
            const Scalar velGradIJ = (slipVelocity - velI) / distance * scvf.directionSign();
 
            flux[scv.dofAxis()] -= (mu * (velGradIJ + velGradJI))*scvf.directionSign();
 
            // advective terms
            if (problem.enableInertiaTerms())
            {
                // transporting velocity corresponds to velJ
                const auto transportingVelocity = [&]()
                {
                    const auto innerTransportingVelocity = elemVolVars[orthogonalScvf.insideScvIdx()].velocity();
 
                    if (!elemVolVars.hasVolVars(orthogonalScvf.outsideScvIdx()))
                        return innerTransportingVelocity; // fallback
 
                    const auto outerTransportingVelocity = elemVolVars[orthogonalScvf.outsideScvIdx()].velocity();
 
                    // if the orthogonal scvf lies on a boundary and if there are outside volvars, we assume that these come from a Dirichlet condition
                    if (orthogonalScvf.boundary())
                        return outerTransportingVelocity;
 
                    static const bool useOldScheme = getParam<bool>("FreeFlow.UseOldTransportingVelocity", true); // TODO how to deprecate?
                    if (useOldScheme)
                        return innerTransportingVelocity;
                    else
                    {
                        // average the transporting velocity by weighting with the scv volumes
                        const auto insideVolume = fvGeometry.scv(orthogonalScvf.insideScvIdx()).volume();
                        const auto outsideVolume = fvGeometry.scv(orthogonalScvf.outsideScvIdx()).volume();
                        const auto outerTransportingVelocity = elemVolVars[orthogonalScvf.outsideScvIdx()].velocity();
                        return (insideVolume*innerTransportingVelocity + outsideVolume*outerTransportingVelocity) / (insideVolume + outsideVolume);
                     }
                }();
 
                // Do not use upwinding here but directly take the slip velocity located on the boundary. Upwinding with a weight of 0.5
                // would actually prevent second order grid convergence.
                const auto rho = problem.insideAndOutsideDensity(fvGeometry.element(), fvGeometry, scvf);
                const auto transportedMomentum = slipVelocity * rho.second;
 
                flux[scv.dofAxis()] += transportingVelocity * transportedMomentum * scvf.directionSign();
            }
        }
        else if (scv.boundary() && problem.onSlipBoundary(fvGeometry, fvGeometry.frontalScvfOnBoundary(scv)))
        {
            /*                                    *
            *                     ---------#######*           x dof position   ***** porous boundary at right side
            *                     |      ||      |*
            *                     |      ||      |*  velI     -- element
            *                     |      || scv  x~~~~>
            *                     |      ||      |*           O position at which gradient is evaluated (integration point)
            *                     | velJ ^       |*^
            *                     -------|-######O*| v_slip   || and # staggered half-control-volume (own element)
            *                     |              |*
            *                     |  neighbor    |*
            *                     |   element     ~~~~> velI outside (Does not exist if lower later lateral scvf lies on non-Dirichlet
            *                     |              |*                  boundary. In that case, use the given tangentialVelocityGradient.)
            *                     |              |*
            *                     ----------------*           :: staggered half-control-volume (neighbor element)
            *                            ^
            *                            |   ^
            *        slip gradient       |   |   ^
            *                            |   |   |
            *
            */
            const Scalar velJ = elemVolVars[orthogonalScvf.insideScvIdx()].velocity();
 
            // viscous terms
            const Scalar mu = problem.effectiveViscosity(fvGeometry.element(), fvGeometry, scvf);
            const Scalar distance = (fvGeometry.scv(orthogonalScvf.insideScvIdx()).dofPosition()- scvf.ipGlobal()).two_norm();
 
            const Scalar velGradIJ = [&]
            {
                if (elemVolVars.hasVolVars(scvf.outsideScvIdx()))
                    return StaggeredVelocityGradients::velocityGradIJ(fvGeometry, scvf, elemVolVars);
                else
                    return tangentialVelocityGradient;
            }();
 
            const Scalar slipVelocity = problem.beaversJosephVelocity(fvGeometry, orthogonalScvf, elemVolVars, velGradIJ)[scvf.normalAxis()]; // TODO rename to slipVelocity
            const Scalar velGradJI = (slipVelocity - velJ) / distance * orthogonalScvf.directionSign();
 
            flux[scv.dofAxis()] -= (mu * (velGradIJ + velGradJI))*scvf.directionSign();
 
            // advective terms
            if (problem.enableInertiaTerms())
            {
                // transporting velocity corresponds to velJ
                const auto transportingVelocity = slipVelocity;
 
                // if the scvf lies on a boundary and if there are outside volvars, we assume that these come from a Dirichlet condition
                if (scvf.boundary() && elemVolVars.hasVolVars(scvf.outsideScvIdx()))
                {
                    flux[scv.dofAxis()] += problem.density(fvGeometry.element(), scv)
                                        * elemVolVars[scvf.outsideScvIdx()].velocity() * transportingVelocity * scvf.directionSign(); // TODO revise density
                    return flux;
                }
 
                const auto innerVelocity = elemVolVars[scvf.insideScvIdx()].velocity();
                const auto outerVelocity = [&]
                {
                    if (!elemVolVars.hasVolVars(scvf.outsideScvIdx()))
                        return innerVelocity; // fallback
                    else
                        return elemVolVars[scvf.outsideScvIdx()].velocity();
                }();
 
                const auto rho = problem.insideAndOutsideDensity(fvGeometry.element(), fvGeometry, scvf);
                const bool selfIsUpstream = scvf.directionSign() == sign(transportingVelocity);
 
                const auto insideMomentum = innerVelocity * rho.first;
                const auto outsideMomentum = outerVelocity * rho.second;
 
                static const auto upwindWeight = getParamFromGroup<Scalar>(problem.paramGroup(), "Flux.UpwindWeight");
 
                const auto transportedMomentum =  selfIsUpstream ? (upwindWeight * insideMomentum + (1.0 - upwindWeight) * outsideMomentum)
                                                                    : (upwindWeight * outsideMomentum + (1.0 - upwindWeight) * insideMomentum);
 
                flux[scv.dofAxis()] += transportingVelocity * transportedMomentum * scvf.directionSign();
            }
        }
 
        return flux;
    }
};
 
} // end namespace Dumux
 
#endif